where \(w \sim v\)  when they are connected by an edge and \(deg_{\Gamma }w\)  is the degree of \(w\) . In the case where the graph is finite (so \(|V|<\infty \) ) \(\Delta \)  may be represented by an \(|V| \times |V|\)  matrix map whose domain is the vectors of the form \((f(v))_{v\in V}\) . The spectral theory of finite graphs aims to study graph properties via the spectrum of this matrix, i.e. the set of its eigenvalues. This theory has numerous applications in theoretical computer science, network theory and spectral geometry.
See [1]} for more details.
